Symmetry-adapted
lattice-dynamical model for single-walled carbon nanotubes (SWNTs)
The phonon
dispersion problem can be made tractable for all tubes of practical interest
(diameters up to 3 nm) if the helical symmetry of the tubes is taken into
account [1,2]. It is easier of all to introduce this
symmetry in comparison with the translational symmetry of graphene. In
graphene, the two-atom unit cell can be mapped unto the entire graphene sheet
by use of two primitive translations. Similarly, the two-atom unit cell of the
nanotube can be mapped unto the entire tube by means of two different screw
operators. A screw operator {S|t} rotates the
position vector of an atom at an angle φ
about the tube axis and translates it at a vector t along the same axis. Let us label the two-atom unit cells by the
vector lattice index by l = (l1,l2), where l1 and l2 are integer numbers, and the atoms in each atomic
pair by the integer κ (κ = 1, 2). Then, the equilibrium
position vector R(lκ) of the κth atom of the lth
unit cell of the tube can be obtained from R(κ) ≡ R(0κ) by means of the two screw operators {S1|t1}
and {S2|t2}:
It is convenient to adopt the compact notation 
and
, and to rewrite this equation as
For small
displacements u(lκ) of the atoms
from their equilibrium positions, the harmonic approximation can be used for
the potential energy of the tube. The resulting Lagrangian is quadratic in the
atomic displacements and velocities. The equation of motion is then readily
derived in the form
Here 
are the force
constants.
The usual
choice of the solutions to the equation of motion as waves along the tube axis
will not reduce the infinite number of equations to a system of finite number
of equation. In order to take advantage of the helical symmetry of the tube,
one can assume solutions of the type
Here, the
eigenvector e is rotated around the
tube axis so as to have the same orientation relative to the tube surface for
all two-atom cells. At this moment, it is convenient to account for the
rotational and translational symmetry of the tubes. Indeed, the displacement
vector u remains unchanged under a
rotation of the system at 2π and
is a wave along the tube axis. The two conditions lead to the following
restrictions on the quantity q: 
and
, where q is the
one-dimensional wavevector of the tube (
) and l is the
azimuthal quantum number (l = 0, 1, 2, …, N−1).
The expression above is then modified to the form
where 
and 
are the dimensionless coordinates of the unit cell around the
circumference and along the tube axis.
Substituting
the expression above in the equation of motion, one easily obtains the matrix
eigenvalue problem of size 6
The
solutions of this system of equations are the phonon eigenvalues 
and eigenvectors
, m = 1, 2, …, 6. This system has to be solved for all values of l (i.e., N times) and as many times as the number of selected values of q.
Therefore, the initial problem for the unit cell of 2N atoms, which needs CPU time increasing
as N3, is reduced to a
linear-N problem in the
symmetry-adapted calculations. The computational advantage of the
symmetry-adapted scheme is obvious.
Presently,
the symmetry-adapted lattice-dynamical model is the only lattice-dynamical
model for calculation of the phonon dispersion of any nanotube of practical
interest. It exists in two parametrizations: a valence-force field version
[1,2] and a force-constant version [3]. Recently, the latter version has been
adopted by Dresselhaus et al.
References
1.
V. N.
Popov, V. E. Van Doren, and M. Balkanski, Phys. Rev. B
59 (1999) 8355-8358.
2.
V. N.
Popov, V. E. Van Doren, and M. Balkanski, Phys. Rev. B
61 (2000) 3078-3084.
3.
Z.
M. Li, V. N. Popov, and Z. K. Tang, Solid State Commun. 130 (2004)
657-661.
Valentin Popov